Abstract
In this paper, a new dynamic model is presented for the experimental data generated by the Madison Symmetric Torus (MST) machine. The model is based on a modified sine-Gordon (SG) dynamic equation. The modified sine-Gordon equation model effectively captures the behavior of the slinky mode in reversed-field pinch experiments. In addition, this paper demonstrates how the derived model accurately describes the behavior of the localized magnetohydrodynamic mode (slinky mode) that appears in reversed-field pinch toroidal magnetic confinement systems. The modified SG equation model is solved analytically by using the perturbation method. The resulting model is fit to match a variety of experimental results in the MST reversed-field pinch experiment. The efficacy of the newly developed model in effectively representing the slinky mode is verified by comparing obtained analytical solution to experimentally measured data.
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Ebraheem, H.K., Shohet, J.L., Scott, A.C.: Mode locking in reversed-field pinch experiments. Phys. Rev. Lett. 88, 235003 (2002)
Shohet, J.L., Barmish, B.R., Ebraheem, H.K., Scott, A.C.: The sine-Gordon equation in reversed-field pinch experiments. Phys. Plasmas 11, 3887 (2004)
Shohet, J.L.: The sine-Gordon equation in toroidal magnetic-fusion experiments. Eur. Phys. J. Special Topics 147, 191–207 (2007)
Yagi, Y., Koguchi, H., Nilsson, J.-A.B., Bolzonella, T., Zanca, P., Sekine, S., Osakabe, K., Sakakita, H.: Phase and wall-locked modes found in a large reversed-field pinch machine. Jpn. J. Appl. Phys. 38, L780 (19990)
Hansen, A.K.: Kinematics of nonlinearly interacting MHD instabilities in a plasma. Ph.D. Thesis, University of Wisconsin -Madison (2000)
Hansen, A.K., Almagri, A.F., Den Hartog, D.J., Prager, S.C., Sarff, J.S.: Locking multiple resonant mode structures in the reversed-field pinch. Phys. Plasmas 5, 2942 (1998)
Hegna, C.C.: Nonlinear tearing mode interactions and mode locking in reversed-field pinches. Phys. Plasmas 3, 4646 (1996)
Fitzpatrick, R.: Formation and locking of the slinky mode in reversed field pinches. Phys. Plasmas 6, 1168 (1999)
Tamano, T., Bard, W.D., Cheng, C., Kondoh, Y., Haye, R.J.L., Lee, P.S., Saito, M., Schaffer, M.J., Taylor, P.L.: Observation of a new toroidally localized kink mode and its role in reverse-field-Pinch plasmas. Phys. Rev. Lett. 59, 1444 (1987)
Sarff, J.S., Assadi, S., Almagri, A.F., Cekic, M., Den Hartog, D.J., Fiksel, G., Hokin, S.A., Ji, H., Prager, S.C., Shen, W., Sidikman, K.L., Stoneking, M.R.: Nonlinear coupling of tearing fluctuations in the Madison Symmetric Torus. Phys. Fluids B 5, 2540 (1993)
Bodin, H.A.B.: Reversed Field Pinch plasma. Nucl. Fusion 30, 1717 (1990)
Ho, Y.L., Prager, S.C., Schnack, D.D.: Nonlinear behavior of the reversed field pinch with nonideal boundary conditions. Phys. Rev. Lett. 62, 1504 (1989)
Almagri, A. F.: The effects of magnetic field errors on reversed-field pinch plasmas, Ph.D. thesis, University of Wisconsin-Madison (1990)
White, R., Fitzpatrick, R.: Effect of rotation and velocity shear on tearing layer stability in tokamak plasmas. Phys. Plasmas 22, (2015)
Fitzpatrick, R.: Phase locking of multi-helicity neoclassical tearing modes in Tokomak plasmas. Phys. Plasmas 22 (2015)
Xu Tao, Hu, Xi-Wei, Qi-Ming, Hu, Qing-Quan, Yu.: Locking of tearing modes by the error field. Chin. Phys. Lett. 22, 9 (2011)
Ivanov, N.V., Kakurin, A.M.: Locking of Small Magnetic Islands by Error Field in T-10 Tokamak. In: 38th EPS Conference on Plasma Physics (2011)
Fitzpatrick, R.: Linear and nonlinear response of a rotating tokomak plasma to a resonant error-field. Phys. Plasmas 21, (2014)
Scott, A.C.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford University Press, Oxford (2006)
Scott, A.C.: Encyclopedia of Nonlinear Science. Taylor and Francis Group, New York (2005)
Remoissenet, M.: Waves Called Solitons, 3rd edn. Springer, Berlin (1999)
Almagri, A.F., Assadi, S., Prager, S.C., Sarff, J.S., Kerst, D.W.: Locked modes and magnetic field errors in the Madison Symmetric Torus. Phys. Fluids B 4, 4080 (1992)
McLaughlin, D.W., Scott, A.C.: Perturbation analysis of fluxon dynamics. Phys. Rev. A 18, 1652 (1978)
Keener, J.P., Mc Laughlin, D.W.: Solitons under perturbations. Phys. Rev. A 16, 777 (1977)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley Inter-Science, New York (1974)
Nayfeh, A.H.: Introduction to Perturbation Techniques. John Wiley & Sons, Hoboken (1981)
Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer, New York (1991)
Ho, Y.L., Prager, S.C.: Stability of a reversed field pinch with resistive and distant boundaries. Phys. Fluids 31, 1673 (1988)
Acknowledgements
The authors would like to thank S. C. Prager, and A. F. Almagri from the MST scientific research group at University of Wisconsin-Madison for providing the experimental data taken at MST.
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Funding was provided by Public Authority of Applied Education and Training (Grant No. TS16-11).
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This work is supported and funded by the Public Authority of Applied Education and Training, Research Project No. (TS-16-11).
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Ebraheem, H.K., Alkhateeb, N.J. & Sultan, E.K. Modeling of the mode dynamics generated by Madison Symmetric Torus machine utilizing a modified sine-Gordon equation. Nonlinear Dyn 93, 1989–2001 (2018). https://doi.org/10.1007/s11071-018-4302-2
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DOI: https://doi.org/10.1007/s11071-018-4302-2